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--- In [log in to unmask], Chris Bates <chris.maths_student@N...>
wrote:

> The following are the axioms
> for a group G:

No, that is the *definition* of a group, it has nothing
to do with axioms.



> Now the exact form a group takes may vary, but if I know something is a
> group I know these things are true. If they were not, if there were some
> groups where there was no identity for example, then the whole of
> current group theory would just collapse.

Quite to the contrary.  If a mathematical construct were
to violate one of the mentioned conditions, it would simply
not be a group.  There is no way of "knowing something is a
group" other than checking whether the definition is met!

The whole building of math is built on the axioms of the
real numbers, which can't be proven, but must be assumed
to be true for the rest of math to work.  Everything else
in math is proven logically on the basis of these axioms.
While the integer and rational numbers are known from the
Real World, the real numbers are more or less our
invention.

One of the axioms is the Axiom of Order.  It claims that
for two arbitrary real numbers x, y exactly one of the
following three statements is true:  x<y, x=y, x>y.  This
appears trivial to us, but isn't if you start from
scratch, as math does.

If it would turn out that the real numbers are not
logically consistent, we'd be in profound feces, but then
somebody would have to explain why our math worked so
flawlessly for the past centuries.


-- Christian Thalmann